Generative modeling of conditional probability distributions on the level-sets of collective variables

Given a probability distribution $\mu$ in $\mathbb{R}^d$ represented by data, we study in this paper the generative modeling of the corresponding conditional probability distributions on the level-sets of a collective variable $\xi:\mathbb{R}^d \rightarrow \mathbb{R}^k$, where $1 \le k<d$. We propose a general and efficient learning approach that can learn generative models on different level-sets of $\xi$ simultaneously. To improve the learning quality on level-sets in low-probability regions, we also propose a data enrichment strategy by utilizing data from enhanced sampling techniques. We demonstrate the effectiveness of our proposed learning approach through concrete numerical examples. The proposed approach is potentially useful for the generative modeling of molecular systems in biophysics.